Trident curve
# Trident curve

In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula:

    $xy+ax^{3}+bx^{2}+cx=d$

Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y = 1, z = 0; if we substitute x = x/z and y = 1/z into the equation of the trident curve, we get

    $ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},$

which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero.

